The paths of Brownian motion is in some sense a random fractal that has a nontrivial geometric structure at all scales. The exact speed of the particle an any time cannot be defined because Brownian motion is not differentiable (only the average speed can be calculated).

Simple explanation

Brownian motion is traditionally explained comparing the particle with the very big ball that is kicked by the team of football players. The remote observer cannot see individual players (molecules) but can see the moving ball that is randomly changing its speed an direction.

Such explanation was first proposed by Albert Einstein. Einstein predicted that Brownian motion of a particle in a fluid at a thermodynamic temperature T is characterized by a diffusion coefficient

where k is Boltzmann's constant and b is the linear drag coefficient on the particle.

Discovered by botanist Robert Brown who was observing pollen grains, this motion is one of the most obvious evidence of the existence of molecules and was an argument in the past debates about the atomic structure of matter.

Random walk

At any time step, the particle receives a random displacement, caused by one or more by molecules hitting it. If its position at time zero is , the position at time n is

,

where the displacements are assumed to be independent.

Some questions that mathematicians try to answer analyzing this model, are:

Does the particle drift to infinity?

Does the particle return to the neighbourhood of the origin infinitely often?

How does the number of "kickers" (molecules) in surrounding impact the speed of the particle?

The serious mathematical analysis of Brownian motion is quite complex and is not presented in this article. Please use references below.